Some generalizations of Oort's conjecture

Abstract

For a prime $p\ge 5$, let ${\mathcal{S}}_g$ be the moduli space over ${\overline{\mathbb{F}}}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\pm 1\}$: (i) certain supersingular Ekedahl--Oort strata when $g$ is even, (ii) the loci in ${\mathcal{S}}_g$ with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when $g\ge 3$, and (iii) the locus in ${\mathcal{S}}_g$ with $a$-number at least $2$ when $g\ge 4$. Consequently, for $g\ge 4$, the complement in ${\mathcal{S}}_g$ of the open locus where the automorphism group is $\{\pm 1\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\ge 5$. We reduce them to statements about affine Deligne--Lusztig varieties for ${\mathrm{GSp}}_{2g}$ and prove analogues of (ii) for ${\mathrm{GL}}_{2g}$ and ${\mathrm{GSO}}_{4m}$.

38 pages